Trin Coll. <1>
Wednesday April 24.
My dear Father
I am much obliged to you for the cheque you sent me. From what Hardcastle <2> tells me, it is quite clear that that announcement you saw in the papers of John Bagwell Esqre <3> having been lately married is a mistake. I may very likely have already got a written down many of the formulæ you are going to send me.
Todhunter <4> says in his book that the old Trigonometrical functions are now quite superseded, and the new Trig-functions universally adopted in England. They were introduced by Professor Peacock. <5> The sine cosine &c are no longer functions of an the an arc but of the ang <illegible deletion> an angle. According to the old definitions the sine cosine &c were lines, and their length depended upon the length of the radius. The sine cosine &c are now not lines but ratios of lines. Such as th <mathematical formula> sin = perpendicular hypotheneuse considering the right angled triangle made by dropping a perpendicular on one of the lines containing any angle. So also <mathematical formula> cosine = base hypotheneuse Tan.= perp base so that the Trigonometrical functions are ratios of one line to an other, and anoth ∴ arithmetical whole numbers or fractions. and invariable for a any given angle however at whatever point in one of the sides lines containing the angle the point P be taken from which we let fall the perpendicular upon the other line which contains the angle. apparently the value of the new method is that, the Trigonometrical functions are thus invariable for any particular Angle. The trigonometrical functions in the old system are are not invariable for any particular arc, but are lengths lines whose length depends upon the value assigned to the radius. Todhunter shows how a formula stated on the old principle may be readily changed into one stated on the new.
Your affect son
Charles
Notes:
1. Trinity College, Cambridge.
2. Henry Hardcastle was also a student at Harrow with Charles.
3. John Bagwell (b.1811), of Marlfield, Ireland, was a barrister and High Sheriff. His wife had lately died, see Doc. No: 08380.
4. Isaac Todhunter (1820–1884), mathematician.
5. Prof George Peacock (1791–1858), mathematician. see Doc. No: 03328, Doc. No: 03351